doubling and halving problem solving year 4

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Doubling and Halving

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How to Use Halving and Doubling for Multiplication

• By Shelley Gray
• June 23, 2020
• No Comments

The halving and doubling strategy for multiplication is one of the most fascinating multiplication strategies. The best part is that it actually IS a strategy – not a trick.

First, let’s take a look at the halving and doubling strategy in action.

To use halving and doubling, you simply  half  one of the factors and  double  the other. Take this example. To solve 25×16, we could double the 25 to make 50 and then half the 16 to make 8. Suddenly this problem becomes much easier to solve!

CONCEPTUAL UNDERSTANDING

It’s essential to know WHY strategies work. Let’s take a look at exactly why the halving and doubling strategy works. It’s easiest to see this in an array.

Here we have a 4×5 array.

Do we still have the same number of objects? Yes! They are just arranged differently. Now we have a 2×10 array. All we did was half the number of rows and double the number of objects in each row.

Try this yourself with 3×8!

DOES IT ALWAYS WORK?

USING DOUBLING AND HALVING

Doubling and halving works best when one of the factors is even, and when the other factor is a number like 5, 10, 25, or 50. For example, doubling and halving would be an efficient strategy for:

TEACHING DOUBLING AND HALVING

Doubling and halving is a fantastic strategy for your students to work with. You can begin modeling doubling and halving with basic facts like 5×6. However, your students must understand WHY it works. It is not enough to simply tell your students that they can double one factor and half the other. The ability to visualize is key to conceptual understanding. This will be best done with arrays, so be sure to include lots of array practice as students build their understanding of this strategy.

Here are some other considerations to make as you teach doubling and halving:

• Allow students to discover this strategy on their own through activities with arrays. You might have students build a 5×6 array, and then a 10×3 array and ask, “What do you notice?” Then build a 4×7 array and a 2×14 array and ask the same question. Encourage students to look for relationships and connections.
• Implement math investigations where you ask pairs of students to investigate questions like:
• Do doubling and halving always work? Prove it!
• For which types of problems does doubling and halving work best?
• What kinds of problems are NOT very well suited to doubling and halving?

RESOURCES TO HELP

See this blog post in video format here.

This next video is intended for upper elementary students and explains the doubling and halving strategy. This can be used as an introduction.

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• In this lesson, we will find double or half of a length

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Doubling and halving

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These activities are designed to teach doubling and halving strategies.

Multiplication and division, AM (Stage 7) 

Prior knowledge

This activty involves using proportional adjustment to solve multiplication problems. Doubling and halving, and trebling and thirding can be used to make multiplication problems easier to solve. For example, 3 x 16 is the same as 6 x 8.  The activity asks students to solve problems, fill in missing numbers in equations using proportional adjustment and solve word problems.

Algebraic notation The essential notation of 2 x  (and 2) for doubling and /2 or  ÷ 2 or ½ as notation for halving. This is then repeated with letters. This not only bridges the gap between “fill in the box” type problems and the x as an unknown number but also introduces students to such notational forms before they are expected to use them. This is essential introductory algebra to build understanding of the language of mathematics. Good discussion is warranted as a follow-up.

Students are also introduced to the concept of proof. It is likely that when students are asked to “explain why doubling and halving always gives the same answer as the original problem” many are likely to write a story. However, the concept of proof and the power of algebra can be followed up in discussions. A teacher led explanation of “what is going on when we play with the symbols using the rules of mathematics we know” should help decode the answer sheet for the problem. An explanation along the lines of “as we don’t know what numbers we actually started with – and just ended up with the same numbers, what we have shown must work for every pair of numbers we can think of…regardless of whether or not the process is actually useful!” should help explain what manipulating the symbols has shown (or proved).

Doubling and halving to find factors Doubling and halving (tripling and thirding etc) is a very useful strategy for finding a full set of factors. However, it does require some idea of prime numbers and how these operate. Start with 1 x n, and double and halve from there. For example 1 x 60 2 x 30 4 x 15 ← look for what goes into 15 20 x 3 ← 3 is a prime so this thread stops, work on the 20 10 x 6 5 x 12 ← other side is now a prime – so stop

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{{item.title}}, my essentials, ask for help, contact edconnect, directory a to z, how to guides, mathematics k–12, doubling and halving: number talk.

Stage 3 – A thinking mathematically targeted teaching opportunity focussed on exploring doubling and halving to solve multiplication problems

Syllabus outcomes and content descriptors from Mathematics K–10 Syllabus © NSW Education Standards Authority (NESA) for and on behalf of the Crown in right of the State of New South Wales, 2023

Collect resources

You will need:

pens or pencils

plain paper, grid paper or dot paper.

Watch Doubling and halving video (8:44).

Transcript of Doubling and halving

[White text on a navy-blue background reads ‘Doubling and halving? A number talk’. Small white text at the bottom reads ‘NSW Mathematics Strategy Professional Learning team (NSWMS PL team). In the bottom right corner, the NSW Government red ‘waratah’ logo.]

Female speaker:

You will need some resources to create a mathematical model. You will need something to draw or write with, some scissors, some sticky tape or glue, and some plain paper, grid paper or dot paper.

[A blue text on white header reads ‘You will need…’ Four bullet points below (as read by speaker). Below, in a still colour image, a sheet of grid paper and another with rows of blue dots, a pencil, a black pen and an orange-handled pair of scissors.]

Hello there, mathematicians. We were doing some problems the other day, and one of those was exploring some of the strategies we could use to solve 25 times 8.

[A sheet of white paper on a light blue background. A person places a small square of paper onto the larger sheet of paper. It has ’25 x 8’ written on it in black pen.]

Some of you may have been working on solving 25 times 18.

[A second square is placed beside the first and has ’25 x 18’ written on it.]

And in both of these problems, we saw the same strategy being used where you double one number and halve the other.

[a rectangle piece of paper is placed at the top of the sheet. It has ‘double or halve’ written on it in black pen.]

So, let's have a look at how Policeman used the strategy double and halving in 25 times 8 to make sense of how this strategy works.

[The person places a ‘LEGO’ policeman figure on the left of the ’25 x 8’ square of paper and removes the ’25 x 18’ piece. A blank square of paper replaces it and is used for the subsequent mathematics.]

Firstly, Policeman used the commutative property of multiplication to rethink the problem as 8 x 25, or 8 25s. Then he halved the 8, which he knows is 4, and then doubled the 25, which he knows is 50. He still wasn't sure what 4 50s was, so he doubled and halved, or halved and doubled, again. So, half of 4 is 2, and then double 50 is 100. Then he could use renaming to know that 2 hundreds is 200.

Now, we know that symbols don't really give us the understanding of what's happening. They're good for recording, but not necessarily good for climbing inside someone's brain. So, let's use an array now to climb inside Policeman's brain.

[On a new sheet of paper, a small rectangle in the centre has rows of purple dots on it. In the upper left, the previous mathematics to reach 200 on the small square of paper.]

Here we have the 8 x 25, which means I have 8 rows, and there's 25 in each row. 8 x 25.

[‘8 x 25’ is written below the rectangle in green marker pen.]

The first thing we saw Policeman do was halve the 8. So, I can cut this array here to show how he halved 8.

[The rectangle of dots is cut lengthways with the orange-handled scissors. The two halves are then placed next to each other.]

And now I've got 4 x 25, and another 4 x 25. And if I move this part of the array up here, may have to slide this down to see if we can fit this in. I think I might even sticky tape this one together. 8 x 25 has been reshaped or reformed into 4 x 50. So, I just accidentally doubled the 25. So cool. And now we can also see that 8 x 25 is equivalent to 4 x 50.

[‘8 x 25 = 4 x 50’ is written in green marker pen in a line below. ]

There's been no new dots added and none have been taken away. So, the array now looks different and we describe it differently, but it still has the exact same number of dots. So, because Policeman wasn't yet confident with knowing and being able to recall a number fact for 4 x 50, he said, “I'll keep doubling and halving”. He halved 4, which is 2 and we can show that here.

[The rectangle of dots is cut once again lengthways and placed side by side. ‘= 2 x 100 = 200’ is added to the line below.]

And then the array reforms into 2 x 100. Wow. Again, we see it doesn't matter how we formed and reformed the array. It didn't change the total number of dots.

[The cut rectangle is replaced by a smaller rectangle filled with rows of tiny dots. A black pen is used to write ‘8’ down the left-hand side and ‘25’ at the bottom.]

Let's see how that plays out with an array, a teeny tiny array, of 8 x 25 that will fit into our frame. So, Policeman started with 8 x 25. And then halved the 8, which reformed it into 4 x 50.

[A second longer and thinner rectangle of dots is placed below the first. A black pen is used to write ‘4’ down the left-hand side and ‘50’ at the bottom.]

And then halved the 4. And when he halved the 4 to 2, it suddenly doubled the number in each row to 100.

[A third even longer and thinner rectangle of dots is placed below the second rectangle. A black pen is used to write ‘2’ down the left-hand side and ‘100’ at the bottom.]

So, you could apply the same thinking if you didn't know, for example, 6 x 8.

[On a blank sheet of white paper a black marker pen is used to write ‘6 x 8’. Further steps explained by speaker as they are written down.]

You could apply it because you could double 6 to get 12 and then halve 8, which is 4, and then double 12, which is 24. And then halve 4, which is 2. And then double 24, which is 48, and then halve 2, which is one. And now you have 48 ones, which is 48.

So, now over to you, mathematicians, to create a model to show how you could use the doubling and halving strategy to solve 6 x 8. You might like to use grid paper or some dot paper, or you could even create a comic strip to show all of the steps of using the doubling and halving strategy to solve 6 x 8.

[A small sheet of grid paper and a small sheet that has rows of red dots on it. A larger sheet has a comic strip of 6 squares with simple black pen drawings in each square.]

Have fun being creative. Over to you, mathematicians.

[White text on a blue background reads ‘Over to you to create a mathematical model to show how you could use doubling and halving to solve 6 x 8.’]

What's some of the mathematics?

[White text on a blue background reads ‘What’s (some of) the mathematics?’]

[A blue text header on a white background reads ‘What’s (some of) the mathematics?’ A bullet point below (as read by speaker) and two of the small squares of hand-written mathematics from earlier.]

When multiplying two numbers, if we double one number and halve the other, the product still remains the same.

[A second bullet point (as read by speaker) and the dissected rectangle of dots from earlier with hand-written mathematics below.]

Reforming or reshaping an array doesn't change the total. This reminds us that quantities like 200 can look different and still have the same value.

[A third bullet point (as read by speaker) and three images below of the two different rows of dots and the comic strip from earlier.]

We can use models and representations to help us make sense of and explain mathematical ideas.

[The NSW Government waratah logo turns briefly in the middle of various circles coloured blue, red, white and black. A copyright symbol and small blue text below it reads ‘State of New South Wales (Department of Education), 2021.’]

[End of transcript]

Instructions

Create a mathematical model to show how you could use doubling and halving to solve 6 x 8. You could use grid paper, dot paper, or even draw a comic strip!

Find 3 examples where doubling and halving isn't an efficient strategy.

Why is it more efficient for some problems and not for others?

Bakery – multiplicative strategies (reSolve)

Stage 3 – A thinking mathematically targeted teaching opportunity focused on developing flexible multiplicative strategies, communicating and reasoning

Colour in fractions

Stage 3 – A thinking mathematically context for practise focussed on deepening understanding of equivalent fractions

Double or halve? – Stage 2 and 3

Stage 2 and 3 – A thinking mathematically context for practise resource focussed on using doubling and halving to reach a target number

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Doubling and Halving Lessons

Subject: Mathematics

Age range: 7-11

Resource type: Lesson (complete)

Last updated

27 March 2021

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A double lesson that focuses on doubling and halving with one lesson on doubling and the second on halving. Each lesson has an arithmetic starter included in the PowerPoint (written for Year 6 but easily changed for other year groups), questions and discussion points and then an activity and apply it/ challenge tasks that involve some form of doubling or halving. Lessons have been created with Year 6 in mind and working on developing understanding but can easily be adapted across Key Stage Two.

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